The backdrop is one of the most celebrated results in quantum field theory in curved spacetime. A detector that falls freely through the Bunch–Davies vacuum of de Sitter spacetime, the maximally symmetric expanding universe, registers a thermal response at the Gibbons–Hawking temperature, which is proportional to the Hubble parameter H divided by 2π. A detector accelerated uniformly through empty Minkowski space registers the Unruh temperature, proportional to its acceleration a divided by 2π. And a detector at rest in a thermal bath at inverse temperature β simply reads the bath temperature. Set a equal to H and choose β equal to 2π/H, and all three local temperatures coincide. The new work pushes this observation to its logical extreme: for the conformal scalar field and the trajectories considered, the complete local Wightman kernels — the full two-point correlation functions a single atom samples — are not merely similar but identical, including their boundary prescriptions.

That local degeneracy means a single two-level atom with energy gap Ω has exactly the same excitation and de-excitation spectrum in all three settings, and even the same response under any common smooth switching of the interaction. The spectrum takes the familiar Planck form, proportional to Ω divided by 2π times one over the exponential factor e raised to 2πΩ/H minus one. In other words, no experiment performed on one atom alone, however carefully timed or switched, can distinguish cosmic expansion from acceleration from heat. The information that separates the three scenarios must live somewhere else — and the paper shows precisely where: in the cross correlations between two separated atoms.

The author, Zhiming Huang, sets up two identical two-level atoms weakly coupled to a four-dimensional massless conformally coupled scalar field. The atoms follow one of three pairs of worldlines: comoving geodesics at fixed comoving coordinates in the spatially flat patch of de Sitter spacetime; transversely separated uniformly accelerated trajectories in the Minkowski vacuum, maintaining a fixed separation L perpendicular to their acceleration; or static trajectories separated by L in a Minkowski thermal Gibbs state. The key mathematical object is the off-diagonal Wightman function, the cross correlation between the field evaluated on one atom’s worldline and on the other’s. Unlike the local kernel, this object depends on the spatial and temporal relationship between the two worldlines, and that dependence turns out to carry an unmistakable fingerprint of the underlying spacetime and state.

At finite separation, the three cross spectra share the same local thermal prefactor but multiply it by different spatial modulation factors. In the thermal bath, the modulation is the sinc factor, sin(ΩL)/(ΩL). In both vacuum settings — de Sitter and accelerated — the modulation is instead a hyperbolic geometric factor, involving the sine of 2Ω divided by the relevant temperature parameter times the inverse hyperbolic sine of half the dimensionless separation, divided by the separation times a square-root factor. When a equals H, the de Sitter and accelerated factors are the same function, but both differ from the thermal sinc. The difference is not cosmetic: for a fixed nonzero atomic gap, the large-distance envelope of the thermal cross spectrum falls off as one over L, while the envelope in the two vacuum settings falls off as one over L squared. A thermal bath keeps whispering between the atoms across large distances in a way that vacuum correlations do not.

The paper also establishes a stronger and more surprising identity. For every prescribed midpoint time T — the average of the two proper times at which the field is sampled on the two de Sitter worldlines — the comoving de Sitter cross kernel equals, exactly, the stationary accelerated kernel with acceleration a equal to H and separation parameter L set equal to r(T), the equal-time physical separation of the comoving pair at that midpoint. Because cosmic expansion stretches physical distances, r(T) grows exponentially with T; the identity therefore says that the expanding-universe correlation at each instant mimics an accelerated-vacuum correlation at a correspondingly larger separation. The equality holds for the entire relative-time dependence of the kernel at that midpoint, and its Fourier transform, a Wigner-type cross spectrum, is then also equal in both the excitation and de-excitation sectors.

Here the paper is unusually careful about interpretation, and the care matters. The fixed-T Fourier transform resolves the two-time kernel at a single midpoint and is explicitly not an instantaneous detector transition rate. The de Sitter side samples pairs of events at times T plus s over two and T minus s over two, spread over all relative times s, while the accelerated side belongs to a genuinely stationary configuration. This distinction becomes physically consequential when finite-time switching enters. A real measurement window samples a range of midpoint times, and since the de Sitter physical separation changes with T, matching the two vacuum kernels at one reference midpoint does not generally make their integrated cross responses equal over the measurement interval. The local responses, by contrast, remain exactly equal for any common smooth switching, because the local kernels agree at all pairs of times. The degeneracy is thus perfect locally and fragile nonlocally.

The analysis also tracks the de-excitation sector, where the Fourier contour picks up different pole contributions in the vacuum and thermal cases, and shows that under the temperature matching all three settings obey the same algebraic relation between negative- and positive-frequency cross spectra, with the ratio given by the Boltzmann factor e raised to βΩ. For the stationary accelerated and thermal configurations these coefficients feed directly into weak-coupling transition rates between the atoms’ collective states, the symmetric and antisymmetric superpositions of one excited and one ground-state atom. For the comoving de Sitter pair, the same relation holds only as an algebraic statement about fixed-midpoint transforms, not as a stationary detailed-balance law governing finite-time dynamics; the author notes that this work extends earlier stationary comparisons to the midpoint-time dependence of a comoving pair.

The scope of the result is bounded, and the paper says so plainly. The kernel identity rests on three technical pillars: conformal coupling, vanishing field mass, and four spacetime dimensions, together with the specific comoving and transversely accelerated trajectories. It is not claimed for massive or minimally coupled fields, nor for arbitrary worldlines, and an appendix on the static de Sitter comparison illustrates how matching a kernel’s spatial parameter must be kept distinct from matching a distance defined on a chosen spatial slice. Within those bounds, however, the comparison is exact and fully analytic, derived by contour integration and mode expansions laid out across the paper’s appendices.

What makes the result conceptually striking is the hierarchy it reveals. At the level of what any single observer’s detector can feel, the expanding universe, the accelerated vacuum, and the hot bath are indistinguishable — the Gibbons–Hawking, Unruh, and thermal effects conspire into one identical Planckian response. At the level of two-atom correlations, the conspiracy dissolves: expansion imprints an exponentially evolving geometric factor, acceleration a stationary hyperbolic one, and heat a slowly decaying sinc. Cross correlations, the same quantity that underlies entanglement harvesting and collective radiative phenomena in earlier detector studies, emerge as the discriminator that single-atom thermometry can never provide. In an era when precision quantum sensors increasingly exploit spatially resolved measurement channels, the message is that the deepest information about a quantum field state may be stored not in what each detector sees, but in what they see together.

Subject of Research: Cross-correlation spectra of two atoms coupled to a scalar field in de Sitter spacetime, the Unruh setting, and a thermal bath

Article Title: Comparison of two-atom cross spectra in de Sitter spacetime, uniformly accelerated Minkowski vacuum, and a thermal bath

Article References: Huang, Z. (2026). Comparison of two-atom cross spectra in de Sitter spacetime, uniformly accelerated Minkowski vacuum, and a thermal bath. The European Physical Journal C, 86(10), Article 1213. https://doi.org/10.1140/epjc/s10052-026-16520-3

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16520-3

Keywords: de Sitter spacetime, Unruh effect, Gibbons–Hawking temperature, quantum field theory in curved spacetime, Wightman function, cross spectrum, detector model, thermal bath, conformal scalar field, entanglement, cosmic expansion, two-level atoms